How to stop cloth-physics from catching on pointy surfaces?
Most of these come with larger simulation times, try to keep the subdivision to a minimum. 


Increase the substeps of the simulation to prevent the geometries from ever intersecting.
Increase the quality of the collision with the Quality slider.
Increase the (minimum) Distance of the collision. Note that this will also create a gap between the object. I would advise to simulate with an offset animated collider, which can be repositioned after baking.
Replace the animated object with simpler geometry. Recreate the basic shape with concave meshes, with evenly distributed triangles while trying to keep the edge angle at a minimum. Hide these helper collision for rendering.